The members of Archaea were most likely the first representatives of cellular life. Prior to that it is hypothesized that self-replicating biomolecule systems existed in "mineral cells" summarized nicely in the article "The rocky roots of the acetyl-CoA pathway" by Russell and Martin (2004).

To my knowledge there is no "first cell" organism that has been discovered....it definitely not fungus...Fungi are eukaryotic.